Here, the prime denotes derivative with respect to \(r\) . In the
presence of the nonlinear BI electrodynamics the Eqs.
(REF ) and (REF ) do not change in comparison with
the linear Maxwell case. In the limiting case where \(b\rightarrow 0\)  the equations of motion of the Maxwell field are reproduced
[1]}. If we consider the probe limit by setting
\(\kappa =0\) , the equations of motion (REF ) and
(REF ) turn to the corresponding equations in
[2]}. There are scaling symmetries of the equations of motion (REF )-(REF ) that we can use to
set \(q\)  and \(l\)  equal to unity.
\(q\rightarrow q/a,\text{ \ \ \ \ }\phi \rightarrow a\phi ,\text{ \ \ \ \ }\rho _{x} \rightarrow a\rho _{x} , \ \ \ \kappa \rightarrow \kappa /a, \ \ \ b\rightarrow b /a^{2},\\\\l\rightarrow al,\text{ \ \ \ \ }r\rightarrow ar,\text{ \ \ \ \ }q\rightarrow q/a, \text{ \ \ \ \ }b\rightarrow b a^{2}, \ \ \ m\rightarrow m/a.\) 
